2.8 problem 8

Internal problem ID [10338]

Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section: Chapter 1, section 1.2. Riccati Equation. 1.2.2. Equations Containing Power Functions
Problem number: 8.
ODE order: 1.
ODE degree: 1.

CAS Maple gives this as type [_Riccati]

\[ \boxed {y^{\prime }-a \,x^{n} y^{2}=b \,x^{m}} \]

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 170

dsolve(diff(y(x),x)=a*x^n*y(x)^2+b*x^m,y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {x^{-\frac {n}{2}+\frac {m}{2}} \sqrt {a b}\, \left (\operatorname {BesselY}\left (\frac {1+m}{n +m +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+\frac {m}{2}+1}}{n +m +2}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {1+m}{n +m +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+\frac {m}{2}+1}}{n +m +2}\right )\right )}{a \left (\operatorname {BesselY}\left (\frac {-n -1}{n +m +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+\frac {m}{2}+1}}{n +m +2}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {-n -1}{n +m +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+\frac {m}{2}+1}}{n +m +2}\right )\right )} \]

✓ Solution by Mathematica

Time used: 2.978 (sec). Leaf size: 1805

DSolve[y'[x]==a*x^n*y[x]^2+b*x^m,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {a^{-\frac {2 m+3 n+5}{2 (m+n+2)}} b^{-\frac {n+1}{2 (m+n+2)}} (m+n+1)^{\frac {n+1}{m+n+2}} \left ((m+n+1)^2\right )^{\frac {n+1}{m+n+2}-\frac {1}{2}} x^{-n-1} \left (x^{m+n+1}\right )^{-\frac {n+1}{2 (m+n+1)}} \left (a^{\frac {n+1}{2 (m+n+2)}} b^{\frac {n+1}{2 (m+n+2)}} (m+n+1)^{-\frac {n+1}{m+n+2}} (m+n+2) \left (-\sqrt {a} \sqrt {b} (m+n+1) \operatorname {BesselJ}\left (\frac {m+1}{m+n+2},\frac {2 \sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}}{\sqrt {(m+n+1)^2} (m+n+2)}\right ) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}+\sqrt {a} \sqrt {b} (m+n+1) \operatorname {BesselJ}\left (-\frac {m+2 n+3}{m+n+2},\frac {2 \sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}}{\sqrt {(m+n+1)^2} (m+n+2)}\right ) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}+(n+1) \sqrt {(m+n+1)^2} \operatorname {BesselJ}\left (-\frac {n+1}{m+n+2},\frac {2 \sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}}{\sqrt {(m+n+1)^2} (m+n+2)}\right )\right ) c_1 \operatorname {Gamma}\left (\frac {m+1}{m+n+2}\right ) \left (x^{m+n+1}\right )^{\frac {n+1}{2 (m+n+1)}}+a^{\frac {n+1}{2 (m+n+2)}} b^{\frac {n+1}{2 (m+n+2)}} (n+1)^2 (m+n+1)^{\frac {n+1}{m+n+2}} \left ((m+n+1)^2\right )^{\frac {m-n}{2 (m+n+2)}} \operatorname {BesselJ}\left (\frac {n+1}{m+n+2},\frac {2 \sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}}{\sqrt {(m+n+1)^2} (m+n+2)}\right ) \operatorname {Gamma}\left (\frac {n+1}{m+n+2}\right ) \left (x^{m+n+1}\right )^{\frac {n+1}{2 (m+n+1)}}+a^{\frac {m+2 n+3}{2 (m+n+2)}} b^{\frac {m+2 n+3}{2 (m+n+2)}} (n+1) (m+n+1)^{\frac {m+2 n+3}{m+n+2}} \left ((m+n+1)^2\right )^{-\frac {n+1}{m+n+2}} \left (\operatorname {BesselJ}\left (-\frac {m+1}{m+n+2},\frac {2 \sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}}{\sqrt {(m+n+1)^2} (m+n+2)}\right )-\operatorname {BesselJ}\left (\frac {n+1}{m+n+2}+1,\frac {2 \sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}}{\sqrt {(m+n+1)^2} (m+n+2)}\right )\right ) \operatorname {Gamma}\left (\frac {n+1}{m+n+2}\right ) \left (x^{m+n+1}\right )^{\frac {m+2 n+3}{2 (m+n+1)}}\right )}{2 \left ((m+n+2) \operatorname {BesselJ}\left (-\frac {n+1}{m+n+2},\frac {2 \sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}}{\sqrt {(m+n+1)^2} (m+n+2)}\right ) c_1 \operatorname {Gamma}\left (\frac {m+1}{m+n+2}\right ) \left ((m+n+1)^2\right )^{\frac {n+1}{m+n+2}}+(n+1) (m+n+1)^{\frac {2 (n+1)}{m+n+2}} \operatorname {BesselJ}\left (\frac {n+1}{m+n+2},\frac {2 \sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}}{\sqrt {(m+n+1)^2} (m+n+2)}\right ) \operatorname {Gamma}\left (\frac {n+1}{m+n+2}\right )\right )} \\ y(x)\to \frac {x^{-n-1} \left (\sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (\frac {1}{m+n+1}+1\right )} \operatorname {BesselJ}\left (\frac {m+1}{m+n+2},\frac {2 \sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}}{\sqrt {(m+n+1)^2} (m+n+2)}\right )-\sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (\frac {1}{m+n+1}+1\right )} \operatorname {BesselJ}\left (-\frac {m+2 n+3}{m+n+2},\frac {2 \sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}}{\sqrt {(m+n+1)^2} (m+n+2)}\right )-(n+1) \sqrt {(m+n+1)^2} \operatorname {BesselJ}\left (-\frac {n+1}{m+n+2},\frac {2 \sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}}{\sqrt {(m+n+1)^2} (m+n+2)}\right )\right )}{2 a \sqrt {(m+n+1)^2} \operatorname {BesselJ}\left (-\frac {n+1}{m+n+2},\frac {2 \sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}}{\sqrt {(m+n+1)^2} (m+n+2)}\right )} \\ y(x)\to \frac {x^{-n-1} \left (\sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (\frac {1}{m+n+1}+1\right )} \operatorname {BesselJ}\left (\frac {m+1}{m+n+2},\frac {2 \sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}}{\sqrt {(m+n+1)^2} (m+n+2)}\right )-\sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (\frac {1}{m+n+1}+1\right )} \operatorname {BesselJ}\left (-\frac {m+2 n+3}{m+n+2},\frac {2 \sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}}{\sqrt {(m+n+1)^2} (m+n+2)}\right )-(n+1) \sqrt {(m+n+1)^2} \operatorname {BesselJ}\left (-\frac {n+1}{m+n+2},\frac {2 \sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}}{\sqrt {(m+n+1)^2} (m+n+2)}\right )\right )}{2 a \sqrt {(m+n+1)^2} \operatorname {BesselJ}\left (-\frac {n+1}{m+n+2},\frac {2 \sqrt {a} \sqrt {b} (m+n+1) \left (x^{m+n+1}\right )^{\frac {1}{2} \left (1+\frac {1}{m+n+1}\right )}}{\sqrt {(m+n+1)^2} (m+n+2)}\right )} \\ \end{align*}